Hypergeometric distribution

An urn contains N balls, of which :

• W are white,
• R are red,

with   W+R = N.

n balls are drawn from the urn, without replacement. That is, n balls (the sample) are randomly selected and taken out of the urn. The number of white balls in the sample is a random variable Y, whose distribution is known as the hypergeometric distribution.
It depends on the three parameters N, W and n and will be denoted HG(N, W, n).

Probability mass function of the hypergeometric distribution

We denote P{Y = w} the probability for the sample to contain exactly w white balls.

with    max(0, n-R)  w  min(B, n).

Mean of the hypergeoemtric distribution

If  p denotes the initial proportion of white balls in the urn, this can also be written :

µ = np

Variance of the hypergeometric distribution

If p denotes the initial proportion of white balls in the urn, this can also be written :

s² = np(1 - p)(1 - (n - 1)/(N - 1))

Note that, for a given value of p, the value of s² converges to np(1 - p), that is, the variance of the binomial distribution with parameters (n, p), when N grows without limit. Can you explain why this should have been expected ?

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These results are demonstrated in the following Tutorial. We call on a technique that is often useful with discrete distribution, and that consists in representing the number of observation in a bin (here, the number of white balls in the sample) as a sum of Bernoulli variables. Other examples of this approach can be found when we study the binomial and multinomial distributions.

THE HYPERGEOMETRIC DISTRIBUTION : BASIC PROPERTIES

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We then address an issue where  the hypergeometric distribution turns up a bit unexpectedly :

    * Let X and Y be two independent binomial random variables, with the same p but different sizes m and n. Choose an integer k, and consider the distribution of X under the condition X + Y = k. This distribution is hypergeometric, and does not depend on p.

This important property of  the binomial distribution is illustrated by a quite instructive interactive animation.

It is the starting point of the Fisher-Irwin test whose purpose is to test the H₀ hypothesis according to which two Bernoulli populations have the same value of the paramter p.
 

DISTRIBUTION OF TWO INDEPENDENT BINOMIAL VARIABLES

CONDITIONALLY TO THEIR SUM

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